and \(P(s, s) = 1 - P(s, s-1) - P(s, s+1)\) .
This is the symmetric random walk with reflecting barrier at \(d\)  and absorbing barrier at 0, with an additional probability to stay put at any vertex. Suppose \(Z_0 = d\)  and \(T = \min \lbrace t\ge 0 : Z_t = 0\rbrace \)  be the first time that \(Z\)  hits 0. Then \({T} \le d^2/2q\) .


Let \(Y\)  be the symmetric random walk on \(\lbrace 0,1,\dots ,2d\rbrace \)  started at \(Y_0 = d\) , with absorbing barriers 0 and \(2d\) .
It is well known - see for example Cox and Miller [1]} - that the expected hitting time of \(\lbrace 0,2d\rbrace \)  by \(Y\)  is \({\min \lbrace t\ge 0 : Y_t \in \lbrace 0,2d\rbrace } = d^2\) .
